
Hence the integrand is a product over the faces \(w\)  of the cell decomposition
of \(\Sigma \) .
We further introduce a local measure on the
discretisation by giving an area \(a_w\)  for every face
\(w\) . The integration factor is a local function \({\Gamma }(\mathcal {U}_w,e^2\,a_w)\) 
depending on the holonomy and area associated to \(w\) .
The correct choice for this local factor which computes the Yang-Mills partition function is [1]}, [2]}
\({\Gamma }\big (\mathcal {U}_w,e^2\,a_w\big ) \operatorname{e}^{-\upsilon _1} \, \sum _{\alpha \in \widehat{G}} \, \dim \alpha \ \chi _\alpha (\mathcal {U}_w) \, \exp \bigg (-e^2\,a_w\, \Big (\frac{C_2(\alpha )}{2} +\upsilon _2\Big ) \bigg ) \ ,\) 
